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I want to do some study about math in the university level, but i have no idea about choosing problems. The fact is that i try to take part in a research activity from our school, but when i got the news,the time has been past five months. Time emergency, I must determining the research question as soon as possible, so Could you give me some small research questions in the university level? I am very grateful! About me: sophomore, Major in Mathematics, except what I have learned in school courses, I have learned some basic number theory, complex analysis by myself.

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I don't quite understand your question. You want to know how to gain experience in research? If that is the question, then you have to talk to the head department of mathematics in your university. He/she will know a lot more about the topics they discuss than anyone here. It really depends on your university. If you are a sophomore then you still have some time to apply for a position but don't wait too long! – diimension Oct 16 '12 at 5:38
Agreeing with diimension. The people in the best position to give you a research question are the people who know you the best, and that's not us, that's the people in the Math Department at your university. – Gerry Myerson Oct 16 '12 at 5:53

Well, haste is usually not a good idea to pick a research problem but it does depend on what is our goal. If your goal is to just do something because everyone else around you is doing something, then okay pick the first decent problem you see. But if your goal is perhaps to have something on your CV/Resume then think a bit carefully. If this is something short term, like just for a semester, then pick whatever. But if this will be long term, like more than a year, or your senior project then pick carefully. If you plan on going to graduate school and getting a Master's or a Ph.D then pick very carefully.

Another distinction about the differences in meaning of the word "research". Undergrads usually doing "research" means like library research where you just learn about a topic and usually present it to an audience. The graduate meaning of "research" is where you actually do something that no one has done before and no one knows the answer. Undergrads do sometimes do original research. Which one are you supposed to do?

If you are doing undergrad "library" research, then you have studied a bit of math, is there something you liked and wanted to study more? I always wanted to learn about number theory which in turn led me into cryptography. I learned C++ which led me to numerical computation in Fortran. I learned a bit about programming which led me into parallel programming. From linear algebra, I wanted to know more about computing matrix exponentials and how to numerically compute eigenvalues and eigenvectors and so on.

If you are talking about original research, then I am afraid you will need to learn a lot more than you may already know. You'll need to know what is known about your problem and then charge ahead with it yourself. It is possible but not easy.

One way of doing both of these is, I just went to the professors (that I liked and I knew I would like to work with now and in the future) and just ask them first what research they do, then tell them what your goal is (library/original research, long-term/short-term) and ask them if they can think of something appropriate for you. You don't have to think of a question yourself so just ask others if they have something they need help with, something appropriate for you.

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I just remembered the "library" research project I did for my complex analysis class. I read up on the Riemann hypothesis and the necessary background. I had always wondered what exactly was it and why is it so important. – Fixed Point Oct 16 '12 at 19:38

The best thing i can told you is ask for the fields of the teachers in the university, but the field of dynamic systems is very interesting and more specific discret dynamic systems for you because as you say you know complex analysis and that is the basic. I recomment you this book for search your question Iteration of Rational Functions: Complex Analytic Dynamical Systems by Alan F. Beardon.

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Lorenz attractors are the stuff of nightmares btw. – Peter Sheldrick Oct 16 '12 at 18:49

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